My experience with cold starts in Azure Functions Serverless is pretty awful. Like most other Azure services, their affordable consumer grade offerings are designed from the ground up not to be good enough for "serious" use.
Cold start times compared to Lambda are worse, and in addition, we would get random 404s which do not appear in any logs; inspecting these 404s indicated they were emitted by nginx, leading me to believe that the ultimate container endpoint was killed for whatever reason but that fact didn't make it back to the router, which attempted and failed to reach the function.
Of course the cold start and 404 are mitigated if you pay for the premium serverless or just host their middleware on their own App Service plans (basically VMs)